Equivalence relation

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Let $S$ be a set. A binary relation $\sim$ on $S$ is said to be an equivalence relation if $\sim$ satisfies the following three properties:

1. For every element $x \in S$, $x \sim x$. (Reflexive property)

2. If $x, y \in S$ such that $x \sim y$, then we also have $y \sim x$. (Symmetric property)

3. If $x, y, z \in S$ such that $x \sim y$ and $y \sim z$, then we also have $x \sim z$. (Transitive property)


Some common examples of equivalence relations:

  • The relation $=$ (equality), on the set of real numbers.
  • The relation $\cong$ (congruence), on the set of geometric figures in the plane.
  • The relation $\sim$ (similarity), on the set of geometric figures in the plane.
  • For a given positive integer $n$, the relation $\equiv \pmod n$, on the set of integers. (Congruence mod n)
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